Transcript University of Washington

The Difference it Makes
Phil Daro
Catching Up
• Students with history of going slower are
not going to catch up without spending
more time and getting more attention.
• Who teaches whom.
• Change the metaphor: not a “gap” but a
knowledge debt and need for know-how.
The knowledge and know-how needed are
concrete, the stepping stones to algebra.
Action
• What is your plan to change the way you
invest student and teacher time?
• What additional resources are you adding to
the base (time)?
• How are you making the teaching of
students who are behind the most exciting
professional work in your district?
System or Sieve?
• A system of interventions that catch
students that need a little help and gives it
• Then catches those that need a little more
and gives it
• Then those who need even more and gives
it
• By layering interventions, minimize the
number who fall through to most expensive
Intensification
Situation of S Needed by S
Intervention
Keeps up
Regular Instruction
None
Struggles some
assignments
Extra feedback on
work, thinking
Classroom Q&A, partner,
teacher’s ear
Not bringing
enough from earlier
lessons each day
Extra support with
regular program
Homework clinic, tutoring,
attention beyond regular
class
Misconceptions
In depth
Sustained instruction with
from earlier grades concentration on
special materials beyond
disrupt participation troublesome concepts regular class
More than a year
behind,
misconceptions
from many years
Intensive ramp-up
course
Designed double period
ramp-up course,
Extended day,
Summer schools
Dylan Wiliam on Instructional
Assessment
Long-cycle
Span: across units, terms
Length: four weeks to one year
Medium-cycle
Span: within and between teaching units
Length: one to four weeks
Short-cycle
Span: within and between lessons
Length:
 day-by-day: 24 to 48 hours
 minute-by-minute: 5 seconds to 2 hours
Strategies for increasing instructional assessment
(Wiliam)
Questioning
Engineering effective classroom
discussions, questions, and learning tasks
Feedback
Moving learners forward with feedback
Sharing Learning
Expectations
Self Assessment
Clarifying and sharing learning intentions
and criteria for success
Peer Assessment
Activating students as instructional
resources for one another
Activating students as the owners of their
own learning
Intensification
Situation of S Needed by S
Intervention
Keeps up
Regular Instruction
None
Struggles some
assignments
Extra feedback on
work, thinking
Classroom Q&A, partner,
teacher’s ear
Not bringing
enough from earlier
lessons each day
Extra support with
regular program
Homework clinic, tutoring,
attention beyond regular
class
Misconceptions
In depth
Sustained instruction with
from earlier grades concentration on
special materials beyond
disrupt participation troublesome concepts regular class
Navigator
More than a year
behind,
misconceptions
from many earlier
grades
Intensive ramp-up
course
Designed double period
ramp-up course,
Extended day,
Summer schools Ramp-Up
or Navigator
Why do students struggle?
•
•
•
•
•
•
Misconceptions
Bugs in procedural knowledge
Mathematics language learning
Meta-cognitive lapses
Lack of knowledge (gaps)
Disposition, belief, and motivation (see
AYD)
Why do students have to do
math. problems?
1. to get answers because
Homeland Security needs them,
pronto
2. I had to, why shouldn’t they?
3. so they will listen in class
4. to learn mathematics
Why give students problems to
solve?
To learn mathematics.
Answers are part of the process, they are not the product.
The product is the student’s mathematical knowledge and
know-how.
The ‘correctness’ of answers is also part of the process.
Yes, an important part.
Wrong Answers
• Are part of the process, too
• What was the student thinking?
• Was it an error of haste or a stubborn
misconception?
Three Responses to a Math
Problem
1. Answer getting
2. Making sense of the problem situation
3. Making sense of the mathematics you can
learn from working on the problem
Answers are a black hole:
hard to escape the pull
• Answer getting short circuits mathematics,
making mathematical sense
• Very habituated in US teachers versus
Japanese teachers
• Devised methods for slowing down,
postponing answer getting
Answer getting vs. learning
mathematics
• USA:
How can I teach my kids to get the answer to this
problem?
Use mathematics they already know. Easy, reliable, works
with bottom half, good for classroom management.
• Japanese:
How can I use this problem to teach mathematics
they don’t already know?
Teaching against the test
3+ 5 =[]
3+[]=8
[]+5=8
8-3 =5
8-5 =3
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Anna bought 3 bags of red gumballs and 5 bags
of white gumballs. Each bag of gumballs had
7 pieces in it. Which expression could Anna use
to find the total number of gumballs she
bought?
A (7 X 3) + 5 =
B (7 X 5) + 3 =
C 7 X (5 + 3) =
D 7 + (5 X 3) =
An input-output table is shown below.
– Input (A) Output (B)
– 7
14
– 12
19
– 20
27
Which of the following could be the rule for the input-output
table?
A. A × 2 = B
B. A + 7 = B
C. A × 5 = B
D. A + 8 = B
SOURCE: Massachusetts Department of Education, Massachusetts
Comprehensive Assessment System, Grade 4, 39, 2006.
Butterfly method
Use butterflies on this TIMSS
item
1/2 + 1/3 +1/4 =
Foil FOIL
• Use the distributive property
• It works for trinomials and polynomials in
general
• What is a polynomial?
• Sum of products = product of sums
• This IS the distributive property when “a” is
a sum
Answer Getting
Getting the answer one way or another and
then stopping
Learning a specific method for solving a
specific kind of problem (100 kinds a year)
Answer Getting Talk
•
•
•
•
Wadja get?
Howdja do it?
Do you remember how to do these?
Here is an easy way to remember how to do
these
• Should you divide or multiply?
• Oh yeah, this is a proportion problem. Let’s
set up a proportion?
Canceling
x5/x2 = x•x• x•x•x / x•x
x5/x5 = x•x• x•x•x / x•x• x•x•x
Misconceptions:
where they come from and how to fix
them
Misconceptions about
misconceptions
• They weren’t listening when they were told
• They have been getting these kinds of
problems wrong from day 1
• They forgot
• The other side in the math wars did this to
the students on purpose
More misconceptions about the
cause of misconceptions
• In the old days, students didn’t make these
mistakes
• They were taught procedures
• They were taught rich problems
• Not enough practice
Maybe
• Teachers’ misconceptions perpetuated to
another generation (where did the teachers
get the misconceptions? How far back does
this go?)
• Mile wide inch deep curriculum causes
haste and waste
• Some concepts are hard to learn
Whatever the Cause
• When students reach your class they are not
blank slates
• They are full of knowledge
• Their knowledge will be flawed and faulty,
half baked and immature; but to them it is
knowledge
• This prior knowledge is an asset and an
interference to new learning
Second grade
• When you add or subtract, line the numbers up on the
right, like this:
• 23
• +9
• Not like this
• 23
• +9
Third Grade
• 3.24 + 2.1 = ?
• If you “Line the numbers up on the right “ like you spent
all last year learning, you get this:
• 3.2 4
• + 2.1
• You get the wrong answer doing what you learned last
year. You don’t know why.
• Teach: line up decimal point.
• Continue developing place value concepts
Fourth and Fifth Grade
• Time to understand the concept of place
value as powers of 10.
• You are lining up the units places, the 10s
places, the 100s places, the tenths places,
the hundredths places
Stubborn Misconceptions
• Misconceptions are often prior knowledge
applied where it does not work
• To the student, it is not a misconception, it
is a concept they learned correctly…
• They don’t know why they are getting the
wrong answer
Research on Retention of Learning: Shell Center: Swan et al
Misconception Learning verses Remedial Learning:
Test Scores
25
20
17.8
15.8
15
19.1
12.7
10
10.4
7.9
5
0
Pr e -te s t
Pos t-te s t
De laye d Te s t
Stude nts w ho w e r e
taught by addr e s sing
m is conce ptions
Stude nts w ho w e r e
taught us ing re m e dial
m e as ur e s
A whole in the head
A whole in the whose head?
+
3/4
+
=
1/3
=
4/7
4/7
The Unit: one
on the Number Line
0
1
2
3
4
Between 0 and 1
0
1/4
3/4
1
2
3
4
Adding on the ruler
0
1/4
2/4
^
1/3
3/4
1^
2/3
1
2
3
Differentiating lesson by lesson
• The arc of the lesson
The arc of the lesson
• Diagnostic: make differences visible; what
are the differences in mathematics that
different students bring to the problem
• All understand the thinking of each: from
least to most mathematically mature
• Converge on grade -level mathematics: pull
students together through the differences in
their thinking
Next lesson
• Start all over again
• Each day brings its differences, they never
go away
Lesson Structure
•
•
•
•
•
•
Pose problem
Start work
Solve problem
Prepare to present
Selected presents
Close
whole class (3-5 min)
solo
(1 min)
pair
(10 min)
pair
(5 min)
whole cls (15 min)
whole cls (5 min)
Posing the problem
• Whole class: pose problem, make sure students
understand the language, no hints at solution
• Focus students on the problem situation, not the
question/answer game. Hide question and ask
them to formulate questions that make situation
into a word problem
• Ask 3-6 questions about the same problem
situation; ramp questions up toward key
mathematics that transfers to other problems
What problem to use?
•
•
•
•
Problems that draw thinking toward the mathematics
you want to teach. NOT too routine, right after learning
how to solve
Ask about a chapter: what is the most important
mathematics students should take with them? Find a
problem that draws attention to this mathematics
Begin chapter with this problem (from lesson 5 thru 10,
or chapter test). This has diagnostic power. Also shows
you where time has to go.
Near end of chapter, external problems needed, e.g.
Shell Centre
Solo-pair work
• Solo honors ‘thinking’ which is solo
• 1 minute is manageable for all, 2 minutes creates
classroom management issues that aren’t worth it.
• An unfinished problem has more mind on it than a
solved problem
• Pairs maximize accountability: no place to hide
• Pairs optimize eartime: everyone is listened to
• You want divergance; diagnostic; make
differences visible
Presentations
• All pairs prepare presentation
• Select 3-5 that show the spread, the differences in
approaches from least to most mature
• Interact with presenters, engage whole class in
questions
• Object and focus is for all to understand thinking
of each, including approaches that didn’t work;
slow presenters down to make thinking explicit
• Go from least to most mature, draw with marker
correspondences across approaches
• Converge on mathematical target of lesson
Close
• Use student work across presentations to
state and explain the key mathematical
ideas of lesson
• Applaud the adaptive problem solving
techniques that come up, the dispositional
behaviors you value, the success in
understanding each others thinking (name
the thought)
The arc of a unit
• Early: diagnostic, organize to make differences
visible
– Pair like students to maximize differences between
pairs
• Middle: spend time where diagnostic lessons show
needs.
• Late: converge on target mathematics
– Pair strong with weak students to minimize differences,
maximize tutoring
Each lesson teaches the whole
chapter
• Each lesson covers 3-4 weeks in 1-2 days
• Lessons build content by
– increasing the resolution of details
– Developing additional technical know-how
– Generalizing range and complexity of problem
situations
– Fitting content into student reasoning
• This is not “spiraling”, this is depth and
thoroughness for durable learning
making sense of math. problems
Word Problem from popular
textbook
• The upper Angel Falls, the highest waterfall
on Earth, are 750 m higher than Niagara
Falls. If each of the falls were 7 m lower,
the upper Angel Falls would be 16 times as
high as Niagara Falls. How high is each
waterfall?
Imagine the Waterfalls: Draw
Diagram it
The Height of Waterfalls
Heights
Height or Waterfalls?
750 m.
Heights we know
750 m.
7 m.
Heights we know and don’t
750 m.
d
d
7 m.
7 m.
Heights we know and don’t
750 m.
d
d
7 m.
Angel = 750 +d + 7
Niagara = d + 7
7 m.
Same height referred to in 2 ways
16d = 750 + d
750 m.
16d
d
d
7 m.
Angel = 750 +d + 7
Niagara = d + 7
7 m.
d=?
16d = 750 + d
15d = 750
d = 50
750 m.
16d
d
Angel = 750 +d + 7
Niagara = d + 7
7 m.
Angel = 750 +50 +7 = 807
Niagara = 50 + 7 = 57
d
7 m.
Activate prior knowledge
• What knowledge?
• “Have you ever seen a waterfall?”
• “What does water look like when it falls?”
What is this problem about?
What is this problem about?
HEIGHT!
Delete “waterfalls” and it does change the
problem at all. Replace waterfall with
flagpoles, buildings, hot air balloons…it
doesn’t matter.
The prior knowledge that needs to be
activated is knowledge of height.
Bad Advice
• “eliminate irrelevant information”
– Before you have made sense of the situation,
how would you know what is relevant?
– After you have made sense, you are already
past the point of worrying about relevance.
What mathematics do we want students to
learn from work on this problem …
• Sasha went 45 miles at 12 mph. How long
did it take?
that they can use on this problem?
– Xavier went 85 miles in two and a half hours.
Going at the same speed, how long would it
take for Xavier to go 12 miles.
Teaching to diagram
• Teaching student to create a diagram about the
relationships of the quantities in the problem that
helps them create a mental mathematical model of
that situation.
• Teach diagramming to one student at a time then
to partners then to larger groups
• As a group
Specific techniques:
• What does that phrase mean? (pointing to a phrase that
refers to a quantity).
• Play the naïve student who doesn’t understand the
situation.
– This is what Harold did is he right?
• Can you show me in a diagram?
– Explain your diagram to me
– Where is (quantity) in your diagram?
– Can you label you diagram?
• What are the quantities and how are they related?
Water Tank
• We are pouring water into a water
tank. 5/6 liter of water is being
poured every 2/3 minute.
– Draw a diagram of this situation
– Make up a question that makes this a
word problem
Test item
• We are pouring water into a water tank. 5/6 liter of
How
many liters of water will have been
poured after one minute?
water is being poured every 2/3 minute.
Where are the numbers going to
come from?
• Not from water tanks. You can change to gas
tanks, swimming pools, or catfish ponds
without changing the meaning of the word
problem.
Numbers:
given, implied or asked about



The number of liters poured
The number of minutes spent pouring
The rate of pouring (which relates liters to
minutes)
Diagrams are reasoning tools
• A diagram should show where each of these
numbers come from. Show liters and show
minutes.
• The diagram should help us reason about the
relationship between liters and minutes in this
situation.
Qu i c k T i m e ™ a n d a
T I F F (Un c o m p re s s e d ) d e c o m p re s s o r
a re n e e d e d t o s e e t h i s p i c t u re .
• The examples range in abstractness. The least
abstract is not a good reasoning tool because
it fails to show where the numbers come
from. The more abstract are easier to reason
with, if the student can make sense of them.
goal
• Our goal is to teach students to make sense
of , produce and reason with abstract
diagrams that show all the numbers, their
relationships.
• A good sense making practice is to first make
a more concrete diagram in early sense
making, then revise it to a more abstract
diagram for reasoning purposes.
• A good teaching practice is to have students
compare and discuss different diagrams for
the same problem.
Word problems are a “genre” of
text
• Read poems differently than we read novels or instructions
or a movie review or a recipe for corn fritters
• Genre are contracts between writer and reader: writer
makes assumptions about how the reader will read; reader
needs to make the right assumptions
• Genre knowledge must be taught and learned
Word problem genre assumptions
• The “context’ of word problems is this:
– I am reading this for math class
• This is about numbers of [who cares what]
• Focus language effort on:
– Where are the numbers coming from in this situation? Domains “
numbers of”, units
– Numbers expressed as phrases in text correspond to mathematical
phrases (expressions)
– The verb is “equals” (+,- are conjunctions)
• Can I find or formulate two different phrases (expressions)
that refer to the same number?
early stages in making sense of
the problem situation
Focus on the domain of meaning from which the numbers
come, what are the quantities in the situation?
Imagine the quantities referred to in the problem by words,
numbers, letters, phrases.
Represent the relationships among quantities in diagrams,
tables, words.
Work out what happens “each time” by increasing x by 1 and
figuring what happens to y; relate each time to the rows of
a table and static diagram.
late stages making sense of
situation
• Imagine and define the domain mathematically, what are
the sets of numbers referred to by the variables?
• Animated understanding: from each time to any time.
Generalized grasp of the mathematical relationship in the
problem situation expressed as an equation; y for any x.
• Equations and graphs
• Can interpret transformed equations (that express x in
terms of y, for example) in terms of the problem situation
Making Sense of the mathematics
in the problem
• How does the table correspond to the graph? Where is this
table row on the graph?
• What does a point (a region) on the graph refer to in the
problem situation?
• How does the equation correspond to the situation (what
do the letters refer to, what do the operations and the = say
about the situation?)?
• How do the equation, graph, table, diagram correspond to
each other feature by feature?
• What kind of equation(s) is it?
Make up a word problem for
which the following equation is
the answer
• y = .03x + 1
What have we learned?
Forget ideologies
• Be skeptical so you can see and hear what is really
going on
• Be pragmatic
• People’s core beliefs are changed by their
experiences and their companionships, not by
authorities or speakers. Be patient with the beliefs
of others …except the belief that intelligence is
fixed; it is not, learning changes intelligence!
Who teaches whom
• Students taught by first year teachers,
substitutes, etc. do not benefit from last
year’s PD
• The worse off a student is, the less likely
they are to be taught by someone who
benefitted from PD
• Change who teaches whom for a really big
effect
Each Day: Teachers solve their
problems in priority (theirs) order
1. Students engaged in some proper activity
2. Teacher impresses students that he/she can
handle being in charge
3. Students respect (make be exchanged for other
positive attitude toward) teacher
4. - Enjoy class (optional)
5. - Work hard
6. - Learn
After 2 or 3 years of teaching
• They have found their way of getting 1, 2, 3 or
else they leave or hideout in the profession
• It’s a bad deal for a teacher to trade 1, 2,3 for a pig
in the poke
• Before they will try your 4,5,6 , You have to offer
a 4,5,6 that either fits their 1,2,3; or you have to
give them an alternative 1,2,3 that seems attractive
to them
• Note: They have to deal with students as persons,
so AYD will appeal to them if it isn’t all on them
1,2,3s that scaffold 4,5,6s that
work
• Major design challenge
• When you hear, “a good program, but hard
to implement” it usually has design
problems of this sort…wrong to blame it on
teacher’s beliefs
Malcolm Swan example:
navigator
• Goldilocks problems that lead to concepts through
work on misconceptions (faulty prior knowledge)
• Discussion craftily scaffolded
• Instructional assessment on all cycles, especially
within lesson
• Tasks easy as possible to engage as activities that
also hook straightaway to questions that lead to
concept
• “encouraged uncertainties” at the door of insights
Pedagogy
• Make conceptions and misconceptions visible to
the student
• Design problems that elicit misconceptions so they
can be dealt with
• Students need to be listened to and responded to
• Partner work
• Revise conceptions
• Debug processes
• Meta-cognitive skills
Diagnostic Learning
•
•
•
•
Revise conceptions
Debug processes
Meta-cognitive learning skills
Social Learning skills: you have to know
how to help each other with math.
homework
Content: Foundations
• Uses simple algebra to repair and strengthen students’
arithmetic foundations:
– Extensive use number line to revisit number concepts
and transfer to knowledge of coordinate graphs
– Explicit use of number properties and properties of
equality in reasoning about arithmetic and transfer to
reasoning with letters and expressions that represent
numbers in algebra
– Use good problems to teach path from concrete
reasoning to symbolic expressions of algebra
• Program takes aim at algebra: anticipates learning
difficulties and prepares students to handle difficulties
• Skills routinely exercised AND non-routine problems
Key Features of a good ramp up
curriculum
• Built from Algebra down, rather than from the
deficiencies up
• Acknowledges that students have unreliable, but
real knowledge about mathematics
• Balance and coherence:
– Skills, problem-solving, and conceptual understanding
with a coherence that revolves around optimizing the
conceptual understanding
What do we mean by conceptual
coherence?
• Apply one important concept in 100 situations
rather than memorizing 100 procedures that do not
transfer to other situations.
– Curriculum is a ‘mile deep’ instead of a ‘mile wide’
– Typical practice is to opt for short-term efficiencies,
rather than teach for general application throughout
mathematics.
– Result: typical students can get B’s on chapter tests, but
don’t remember what they ‘learned’ later when they
need to learn more mathematics
– Use basic “rules of arithmetic” (same as algebra)
instead of clutter of specific named methods
Teach from misconceptions
• Most common misconceptions consist of applying
a correctly learned procedure to an inappropriate
situation.
• Lessons are designed to surface and deal with the
most common misconceptions
• Create ‘cognitive conflict’ to help students revise
misconceptions
– Misconceptions interfere with initial teaching and that’s
why repeated initial teaching does’t work
Key features of the a well
designed intervention
• Lean and clean lessons that are simple and focused on the
math to be learned
• Rituals and Routines that maximize student interaction
with the mathematics
• Emphasis on students, student work, and student discourse
• Teaches and motivates how to be a good math student
• Assessment that is ongoing and instrumental in promoting
student learning
• Support for teachers
– The mathematics
– Student work with commentary, and guidance on getting the
full power of the mathematics from the workshop
Bottom up
Schools fill interventions from the bottom up,
not from Algebra down.
Ready for Algebra
1 double-period year away from ready
area, fractions with like denominators, single digit addition and multiplication
facts, read at 4th grade level, …
More than 1 double-period year away
Plan for it
• Need a plan from the bottom up so students
get what they need
Malcolm Swan example
• Goldilocks problems that lead to concepts through
work on misconceptions (faulty prior knowledge)
• Discussion craftily scaffolded
• Instructional assessment on all cycles, especially
within lesson
• Tasks easy as possible to engage as activities that
also hook straightaway to questions that lead to
concept
• “encouraged uncertainties” at the door of insights
Tiered Levels of Intervention
Situation of
Student
Needed by
Student
Interventi
on Tier
Keps up
Regular Instruction
NA
Struggles some
assignments
Extra feedback on
work, thinking
Not bringing
enough from earlier
lessons each day
Extra support with
regular program
Misconceptions
disrupt
participation and
success in
mathematics
(gaps )
In depth
concentration on
troublesome
concepts (not initial
teaching)
Tier 2
More than a year
behind,
misconceptions
from many years
Extra time and focus
on critical
mathematics to
accelerate to grade
Tier 3
Tier 1
Intervention
None
Classroom Q&A, partner,
teacher’s ear
Professional development
Homework clinic, tutoring,
attention beyond regular class
Scheduling / targeted use of
adopted materials
Sustained instruction with
special materials beyond
regular class period and/ or
summer school
Navigator
Designed double period
ramp-up course, summer
school:
Navigator Summer,
Student Materials, Classroom Routines, and Tasks
About Tasks: Interpreting
Multiple Representations
Student Materials, Classroom Routines, and Tasks
About Tasks:
Making Posters
Social and meta-cognitive skills
have to be taught by design
• Beliefs about one’s own mathematical intelligence
– “good at math” vs. learning makes me smarter
• Meta-cognitive engagement modeled and
prompted
– Does this make sense?
– What did I do wrong?
• Social skills: learning how to help and be helped
with math work => basic skill for algebra: do
homework together, study for test together
Marita’s homework
Diagnostic Teaching
• Goal is to surface and make students aware of their
misconceptions
•Begin with a problem or activity that surfaces the
various ways students may think about the math.
•Engage in reflective discussion (challenging for
teachers but research shows that it develops long-term
learning)
•Reference: Bell, A. Principles for the Design of Teaching
Educational Studies in Mathematics. 24: 5-34, 1993
Operations and Word Problems
Operations and Word Problems
Operations and Word Problems
Knowing Fractions
Knowing Fractions
Knowing Fractions
Knowing Fractions
Understanding Fractions
Understanding Fractions